On the upper chromatic index of a multigraph - Mario Gionfriddo; Lorenzo Milazzo; Vitaly Voloshin (Computer Science Journal of Moldova, 2002)
We consider the colorings of the edges of a multigraph in such a way that every non-pendant vertex is incident to at least two edges of the same color. The maximum number of colors that can be used in such colorings is the upper chromatic index of a multigraph G, denoted by not(χ')(G). We prove that if a multigraph G has n vertices, m edges, p pendant vertices and maximum number c disjoint cycles, then not(χ')(G) = c + m - n + p.